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Full text of "Treatise On Analysis Vol-Ii"

10   SPACES WITH OPERATORS. ORBIT SPACES       47

If E is a Banach space, the group GL(E) of linear homeomorphisms of E
onto E (12.8.1) acts continuously on E by (u, x)i~+u(x) (5.7.4).

Let G be a topological group and E a topological space. The mapping
(s9 x)i-+x of G x E into E is a continuous action of G on E, called the
trivial action.

Let G be a topological group acting continuously on a topological space E.
If p is a continuous homomorphism of a topological group G' into G, then G'
acts continuously on E by (s'y x)\~*p(sf) - x.

(12.10.3) If G is a topological group acting continuously on a topological
space E, then for each s e G the mapping xt-*s - x is a homeomorphism of E
onto E.

For it is a continuous bijection, and the inverse bijection x^^s"1  xis
also continuous.

(12.10.4)    If G is a topological group acting continuously on a Hausdorff
topological space E, the stabilizer of each point ofE is a closed subgroup ofG.

This follows immediately from (12.3.5).

(12.10.5)    Let G be a metrizable group acting continuously on a metrizable
space E. Let A be a compact subset of G, and B a closed (resp. compact)
subset ofE. Then A  B is closed (resp. compact) in E.

The second assertion follows from the fact that A  B is the image of the
compact set A x B (3.20.1 6(v)) by the continuous mapping (s,x)h-*s-x
(3.17.9). As to the first assertion, consider a sequence (sn  xn) of points of
A  B (where sn e A and xn e B) with a limit z e E. By hypothesis, the se-
quence (sn) has a subsequence (snk) converging to some point a e A. Since
xnk = Snkl - (snk  xttk), the sequence (xnj) converges to a""1  z. But B is closed
in E and therefore a""1  z e B, hence z = a - (a""1  z) e A  B. By (3.13.13),
the proof is complete.

Let G be a topological group acting continuously on a topological space
E. Let E/G be the set of orbits and let TC : E -> E/G be the canonical mapping,
so that n(x) is the orbit G - x for each x e E. Let C be the set of subsets U of
E/G such that TiT^U) is open in E. It follows immediately from the formulas
(1.8.5) and (18,6) that D is a topology on E/G. The set E/G, endowed with G, the product group H x K acts continuously